/*
 *  Elliptic curves over GF(p): generic functions
 *
 *  Copyright (C) 2006-2015, ARM Limited, All Rights Reserved
 *  SPDX-License-Identifier: GPL-2.0
 *
 *  This program is free software; you can redistribute it and/or modify
 *  it under the terms of the GNU General Public License as published by
 *  the Free Software Foundation; either version 2 of the License, or
 *  (at your option) any later version.
 *
 *  This program is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU General Public License for more details.
 *
 *  You should have received a copy of the GNU General Public License along
 *  with this program; if not, write to the Free Software Foundation, Inc.,
 *  51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
 *
 *  This file is part of mbed TLS (https://tls.mbed.org)
 */

/*
 * References:
 *
 * SEC1 http://www.secg.org/index.php?action=secg,docs_secg
 * GECC = Guide to Elliptic Curve Cryptography - Hankerson, Menezes, Vanstone
 * FIPS 186-3 http://csrc.nist.gov/publications/fips/fips186-3/fips_186-3.pdf
 * RFC 4492 for the related TLS structures and constants
 * RFC 7748 for the Curve448 and Curve25519 curve definitions
 *
 * [Curve25519] http://cr.yp.to/ecdh/curve25519-20060209.pdf
 *
 * [2] CORON, Jean-S'ebastien. Resistance against differential power analysis
 *     for elliptic curve cryptosystems. In : Cryptographic Hardware and
 *     Embedded Systems. Springer Berlin Heidelberg, 1999. p. 292-302.
 *     <http://link.springer.com/chapter/10.1007/3-540-48059-5_25>
 *
 * [3] HEDABOU, Mustapha, PINEL, Pierre, et B'EN'ETEAU, Lucien. A comb method to
 *     render ECC resistant against Side Channel Attacks. IACR Cryptology
 *     ePrint Archive, 2004, vol. 2004, p. 342.
 *     <http://eprint.iacr.org/2004/342.pdf>
 */

#if !defined(MBEDTLS_CONFIG_FILE)
#include "mbedtls/config.h"
#else
#include MBEDTLS_CONFIG_FILE
#endif

#if defined(MBEDTLS_ECP_C)

#include "mbedtls/ecp.h"
#include "mbedtls/threading.h"
#include "mbedtls/platform_util.h"

#include <string.h>

#if !defined(MBEDTLS_ECP_ALT)

#if defined(MBEDTLS_PLATFORM_C)
#include "mbedtls/platform.h"
#else
#include <stdlib.h>
#include <stdio.h>
#define mbedtls_printf     printf
#define mbedtls_calloc    calloc
#define mbedtls_free       free
#endif

#include "mbedtls/ecp_internal.h"

#if ( defined(__ARMCC_VERSION) || defined(_MSC_VER) ) && \
    !defined(inline) && !defined(__cplusplus)
#define inline __inline
#endif

#if defined(MBEDTLS_SELF_TEST)
/*
 * Counts of point addition and doubling, and field multiplications.
 * Used to test resistance of point multiplication to simple timing attacks.
 */
static unsigned long add_count, dbl_count, mul_count;
#endif

#if defined(MBEDTLS_ECP_DP_SECP192R1_ENABLED) ||   \
    defined(MBEDTLS_ECP_DP_SECP224R1_ENABLED) ||   \
    defined(MBEDTLS_ECP_DP_SECP256R1_ENABLED) ||   \
    defined(MBEDTLS_ECP_DP_SECP384R1_ENABLED) ||   \
    defined(MBEDTLS_ECP_DP_SECP521R1_ENABLED) ||   \
    defined(MBEDTLS_ECP_DP_BP256R1_ENABLED)   ||   \
    defined(MBEDTLS_ECP_DP_BP384R1_ENABLED)   ||   \
    defined(MBEDTLS_ECP_DP_BP512R1_ENABLED)   ||   \
    defined(MBEDTLS_ECP_DP_SECP192K1_ENABLED) ||   \
    defined(MBEDTLS_ECP_DP_SECP224K1_ENABLED) ||   \
    defined(MBEDTLS_ECP_DP_SECP256K1_ENABLED)
#define ECP_SHORTWEIERSTRASS
#endif

#if defined(MBEDTLS_ECP_DP_CURVE25519_ENABLED) || \
    defined(MBEDTLS_ECP_DP_CURVE448_ENABLED)
#define ECP_MONTGOMERY
#endif

/*
 * Curve types: internal for now, might be exposed later
 */
typedef enum {
    ECP_TYPE_NONE = 0,
    ECP_TYPE_SHORT_WEIERSTRASS,    /* y^2 = x^3 + a x + b      */
    ECP_TYPE_MONTGOMERY,           /* y^2 = x^3 + a x^2 + x    */
} ecp_curve_type;

/*
 * List of supported curves:
 *  - internal ID
 *  - TLS NamedCurve ID (RFC 4492 sec. 5.1.1, RFC 7071 sec. 2)
 *  - size in bits
 *  - readable name
 *
 * Curves are listed in order: largest curves first, and for a given size,
 * fastest curves first. This provides the default order for the SSL module.
 *
 * Reminder: update profiles in x509_crt.c when adding a new curves!
 */
static const mbedtls_ecp_curve_info ecp_supported_curves[] = {
#if defined(MBEDTLS_ECP_DP_SECP521R1_ENABLED)
    { MBEDTLS_ECP_DP_SECP521R1,    25,     521,    "secp521r1"         },
#endif
#if defined(MBEDTLS_ECP_DP_BP512R1_ENABLED)
    { MBEDTLS_ECP_DP_BP512R1,      28,     512,    "brainpoolP512r1"   },
#endif
#if defined(MBEDTLS_ECP_DP_SECP384R1_ENABLED)
    { MBEDTLS_ECP_DP_SECP384R1,    24,     384,    "secp384r1"         },
#endif
#if defined(MBEDTLS_ECP_DP_BP384R1_ENABLED)
    { MBEDTLS_ECP_DP_BP384R1,      27,     384,    "brainpoolP384r1"   },
#endif
#if defined(MBEDTLS_ECP_DP_SECP256R1_ENABLED)
    { MBEDTLS_ECP_DP_SECP256R1,    23,     256,    "secp256r1"         },
#endif
#if defined(MBEDTLS_ECP_DP_SECP256K1_ENABLED)
    { MBEDTLS_ECP_DP_SECP256K1,    22,     256,    "secp256k1"         },
#endif
#if defined(MBEDTLS_ECP_DP_BP256R1_ENABLED)
    { MBEDTLS_ECP_DP_BP256R1,      26,     256,    "brainpoolP256r1"   },
#endif
#if defined(MBEDTLS_ECP_DP_SECP224R1_ENABLED)
    { MBEDTLS_ECP_DP_SECP224R1,    21,     224,    "secp224r1"         },
#endif
#if defined(MBEDTLS_ECP_DP_SECP224K1_ENABLED)
    { MBEDTLS_ECP_DP_SECP224K1,    20,     224,    "secp224k1"         },
#endif
#if defined(MBEDTLS_ECP_DP_SECP192R1_ENABLED)
    { MBEDTLS_ECP_DP_SECP192R1,    19,     192,    "secp192r1"         },
#endif
#if defined(MBEDTLS_ECP_DP_SECP192K1_ENABLED)
    { MBEDTLS_ECP_DP_SECP192K1,    18,     192,    "secp192k1"         },
#endif
    { MBEDTLS_ECP_DP_NONE,          0,     0,      NULL                },
};

#define ECP_NB_CURVES   sizeof( ecp_supported_curves ) /    \
    sizeof( ecp_supported_curves[0] )

static mbedtls_ecp_group_id ecp_supported_grp_id[ECP_NB_CURVES];

/*
 * List of supported curves and associated info
 */
const mbedtls_ecp_curve_info *mbedtls_ecp_curve_list(void) {
    return (ecp_supported_curves);
}

/*
 * List of supported curves, group ID only
 */
const mbedtls_ecp_group_id *mbedtls_ecp_grp_id_list(void) {
    static int init_done = 0;

    if (! init_done) {
        size_t i = 0;
        const mbedtls_ecp_curve_info *curve_info;

        for (curve_info = mbedtls_ecp_curve_list();
                curve_info->grp_id != MBEDTLS_ECP_DP_NONE;
                curve_info++) {
            ecp_supported_grp_id[i++] = curve_info->grp_id;
        }
        ecp_supported_grp_id[i] = MBEDTLS_ECP_DP_NONE;

        init_done = 1;
    }

    return (ecp_supported_grp_id);
}

/*
 * Get the curve info for the internal identifier
 */
const mbedtls_ecp_curve_info *mbedtls_ecp_curve_info_from_grp_id(mbedtls_ecp_group_id grp_id) {
    const mbedtls_ecp_curve_info *curve_info;

    for (curve_info = mbedtls_ecp_curve_list();
            curve_info->grp_id != MBEDTLS_ECP_DP_NONE;
            curve_info++) {
        if (curve_info->grp_id == grp_id)
            return (curve_info);
    }

    return (NULL);
}

/*
 * Get the curve info from the TLS identifier
 */
const mbedtls_ecp_curve_info *mbedtls_ecp_curve_info_from_tls_id(uint16_t tls_id) {
    const mbedtls_ecp_curve_info *curve_info;

    for (curve_info = mbedtls_ecp_curve_list();
            curve_info->grp_id != MBEDTLS_ECP_DP_NONE;
            curve_info++) {
        if (curve_info->tls_id == tls_id)
            return (curve_info);
    }

    return (NULL);
}

/*
 * Get the curve info from the name
 */
const mbedtls_ecp_curve_info *mbedtls_ecp_curve_info_from_name(const char *name) {
    const mbedtls_ecp_curve_info *curve_info;

    for (curve_info = mbedtls_ecp_curve_list();
            curve_info->grp_id != MBEDTLS_ECP_DP_NONE;
            curve_info++) {
        if (strcmp(curve_info->name, name) == 0)
            return (curve_info);
    }

    return (NULL);
}

/*
 * Get the type of a curve
 */
static inline ecp_curve_type ecp_get_type(const mbedtls_ecp_group *grp) {
    if (grp->G.X.p == NULL)
        return (ECP_TYPE_NONE);

    if (grp->G.Y.p == NULL)
        return (ECP_TYPE_MONTGOMERY);
    else
        return (ECP_TYPE_SHORT_WEIERSTRASS);
}

/*
 * Initialize (the components of) a point
 */
void mbedtls_ecp_point_init(mbedtls_ecp_point *pt) {
    if (pt == NULL)
        return;

    mbedtls_mpi_init(&pt->X);
    mbedtls_mpi_init(&pt->Y);
    mbedtls_mpi_init(&pt->Z);
}

/*
 * Initialize (the components of) a group
 */
void mbedtls_ecp_group_init(mbedtls_ecp_group *grp) {
    if (grp == NULL)
        return;

    memset(grp, 0, sizeof(mbedtls_ecp_group));
}

/*
 * Initialize (the components of) a key pair
 */
void mbedtls_ecp_keypair_init(mbedtls_ecp_keypair *key) {
    if (key == NULL)
        return;

    mbedtls_ecp_group_init(&key->grp);
    mbedtls_mpi_init(&key->d);
    mbedtls_ecp_point_init(&key->Q);
}

/*
 * Unallocate (the components of) a point
 */
void mbedtls_ecp_point_free(mbedtls_ecp_point *pt) {
    if (pt == NULL)
        return;

    mbedtls_mpi_free(&(pt->X));
    mbedtls_mpi_free(&(pt->Y));
    mbedtls_mpi_free(&(pt->Z));
}

/*
 * Unallocate (the components of) a group
 */
void mbedtls_ecp_group_free(mbedtls_ecp_group *grp) {
    size_t i;

    if (grp == NULL)
        return;

    if (grp->h != 1) {
        mbedtls_mpi_free(&grp->P);
        mbedtls_mpi_free(&grp->A);
        mbedtls_mpi_free(&grp->B);
        mbedtls_ecp_point_free(&grp->G);
        mbedtls_mpi_free(&grp->N);
    }

    if (grp->T != NULL) {
        for (i = 0; i < grp->T_size; i++)
            mbedtls_ecp_point_free(&grp->T[i]);
        mbedtls_free(grp->T);
    }

    mbedtls_platform_zeroize(grp, sizeof(mbedtls_ecp_group));
}

/*
 * Unallocate (the components of) a key pair
 */
void mbedtls_ecp_keypair_free(mbedtls_ecp_keypair *key) {
    if (key == NULL)
        return;

    mbedtls_ecp_group_free(&key->grp);
    mbedtls_mpi_free(&key->d);
    mbedtls_ecp_point_free(&key->Q);
}

/*
 * Copy the contents of a point
 */
int mbedtls_ecp_copy(mbedtls_ecp_point *P, const mbedtls_ecp_point *Q) {
    int ret;

    MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&P->X, &Q->X));
    MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&P->Y, &Q->Y));
    MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&P->Z, &Q->Z));

cleanup:
    return (ret);
}

/*
 * Copy the contents of a group object
 */
int mbedtls_ecp_group_copy(mbedtls_ecp_group *dst, const mbedtls_ecp_group *src) {
    return mbedtls_ecp_group_load(dst, src->id);
}

/*
 * Set point to zero
 */
int mbedtls_ecp_set_zero(mbedtls_ecp_point *pt) {
    int ret;

    MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&pt->X, 1));
    MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&pt->Y, 1));
    MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&pt->Z, 0));

cleanup:
    return (ret);
}

/*
 * Tell if a point is zero
 */
int mbedtls_ecp_is_zero(mbedtls_ecp_point *pt) {
    return (mbedtls_mpi_cmp_int(&pt->Z, 0) == 0);
}

/*
 * Compare two points lazyly
 */
int mbedtls_ecp_point_cmp(const mbedtls_ecp_point *P,
                          const mbedtls_ecp_point *Q) {
    if (mbedtls_mpi_cmp_mpi(&P->X, &Q->X) == 0 &&
            mbedtls_mpi_cmp_mpi(&P->Y, &Q->Y) == 0 &&
            mbedtls_mpi_cmp_mpi(&P->Z, &Q->Z) == 0) {
        return (0);
    }

    return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);
}

/*
 * Import a non-zero point from ASCII strings
 */
int mbedtls_ecp_point_read_string(mbedtls_ecp_point *P, int radix,
                                  const char *x, const char *y) {
    int ret;

    MBEDTLS_MPI_CHK(mbedtls_mpi_read_string(&P->X, radix, x));
    MBEDTLS_MPI_CHK(mbedtls_mpi_read_string(&P->Y, radix, y));
    MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&P->Z, 1));

cleanup:
    return (ret);
}

/*
 * Export a point into unsigned binary data (SEC1 2.3.3)
 */
int mbedtls_ecp_point_write_binary(const mbedtls_ecp_group *grp, const mbedtls_ecp_point *P,
                                   int format, size_t *olen,
                                   unsigned char *buf, size_t buflen) {
    int ret = 0;
    size_t plen;

    if (format != MBEDTLS_ECP_PF_UNCOMPRESSED &&
            format != MBEDTLS_ECP_PF_COMPRESSED)
        return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);

    /*
     * Common case: P == 0
     */
    if (mbedtls_mpi_cmp_int(&P->Z, 0) == 0) {
        if (buflen < 1)
            return (MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL);

        buf[0] = 0x00;
        *olen = 1;

        return (0);
    }

    plen = mbedtls_mpi_size(&grp->P);

    if (format == MBEDTLS_ECP_PF_UNCOMPRESSED) {
        *olen = 2 * plen + 1;

        if (buflen < *olen)
            return (MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL);

        buf[0] = 0x04;
        MBEDTLS_MPI_CHK(mbedtls_mpi_write_binary(&P->X, buf + 1, plen));
        MBEDTLS_MPI_CHK(mbedtls_mpi_write_binary(&P->Y, buf + 1 + plen, plen));
    } else if (format == MBEDTLS_ECP_PF_COMPRESSED) {
        *olen = plen + 1;

        if (buflen < *olen)
            return (MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL);

        buf[0] = 0x02 + mbedtls_mpi_get_bit(&P->Y, 0);
        MBEDTLS_MPI_CHK(mbedtls_mpi_write_binary(&P->X, buf + 1, plen));
    }

cleanup:
    return (ret);
}

/*
 * Import a point from unsigned binary data (SEC1 2.3.4)
 */
int mbedtls_ecp_point_read_binary(const mbedtls_ecp_group *grp, mbedtls_ecp_point *pt,
                                  const unsigned char *buf, size_t ilen) {
    int ret;
    size_t plen;

    if (ilen < 1)
        return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);

    if (buf[0] == 0x00) {
        if (ilen == 1)
            return (mbedtls_ecp_set_zero(pt));
        else
            return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);
    }

    plen = mbedtls_mpi_size(&grp->P);

    if (buf[0] != 0x04)
        return (MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE);

    if (ilen != 2 * plen + 1)
        return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);

    MBEDTLS_MPI_CHK(mbedtls_mpi_read_binary(&pt->X, buf + 1, plen));
    MBEDTLS_MPI_CHK(mbedtls_mpi_read_binary(&pt->Y, buf + 1 + plen, plen));
    MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&pt->Z, 1));

cleanup:
    return (ret);
}

/*
 * Import a point from a TLS ECPoint record (RFC 4492)
 *      struct {
 *          opaque point <1..2^8-1>;
 *      } ECPoint;
 */
int mbedtls_ecp_tls_read_point(const mbedtls_ecp_group *grp, mbedtls_ecp_point *pt,
                               const unsigned char **buf, size_t buf_len) {
    unsigned char data_len;
    const unsigned char *buf_start;

    /*
     * We must have at least two bytes (1 for length, at least one for data)
     */
    if (buf_len < 2)
        return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);

    data_len = *(*buf)++;
    if (data_len < 1 || data_len > buf_len - 1)
        return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);

    /*
     * Save buffer start for read_binary and update buf
     */
    buf_start = *buf;
    *buf += data_len;

    return mbedtls_ecp_point_read_binary(grp, pt, buf_start, data_len);
}

/*
 * Export a point as a TLS ECPoint record (RFC 4492)
 *      struct {
 *          opaque point <1..2^8-1>;
 *      } ECPoint;
 */
int mbedtls_ecp_tls_write_point(const mbedtls_ecp_group *grp, const mbedtls_ecp_point *pt,
                                int format, size_t *olen,
                                unsigned char *buf, size_t blen) {
    int ret;

    /*
     * buffer length must be at least one, for our length byte
     */
    if (blen < 1)
        return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);

    if ((ret = mbedtls_ecp_point_write_binary(grp, pt, format,
                                              olen, buf + 1, blen - 1)) != 0)
        return (ret);

    /*
     * write length to the first byte and update total length
     */
    buf[0] = (unsigned char) * olen;
    ++*olen;

    return (0);
}

/*
 * Set a group from an ECParameters record (RFC 4492)
 */
int mbedtls_ecp_tls_read_group(mbedtls_ecp_group *grp, const unsigned char **buf, size_t len) {
    uint16_t tls_id;
    const mbedtls_ecp_curve_info *curve_info;

    /*
     * We expect at least three bytes (see below)
     */
    if (len < 3)
        return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);

    /*
     * First byte is curve_type; only named_curve is handled
     */
    if (*(*buf)++ != MBEDTLS_ECP_TLS_NAMED_CURVE)
        return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);

    /*
     * Next two bytes are the namedcurve value
     */
    tls_id = *(*buf)++;
    tls_id <<= 8;
    tls_id |= *(*buf)++;

    if ((curve_info = mbedtls_ecp_curve_info_from_tls_id(tls_id)) == NULL)
        return (MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE);

    return mbedtls_ecp_group_load(grp, curve_info->grp_id);
}

/*
 * Write the ECParameters record corresponding to a group (RFC 4492)
 */
int mbedtls_ecp_tls_write_group(const mbedtls_ecp_group *grp, size_t *olen,
                                unsigned char *buf, size_t blen) {
    const mbedtls_ecp_curve_info *curve_info;

    if ((curve_info = mbedtls_ecp_curve_info_from_grp_id(grp->id)) == NULL)
        return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);

    /*
     * We are going to write 3 bytes (see below)
     */
    *olen = 3;
    if (blen < *olen)
        return (MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL);

    /*
     * First byte is curve_type, always named_curve
     */
    *buf++ = MBEDTLS_ECP_TLS_NAMED_CURVE;

    /*
     * Next two bytes are the namedcurve value
     */
    buf[0] = curve_info->tls_id >> 8;
    buf[1] = curve_info->tls_id & 0xFF;

    return (0);
}

/*
 * Wrapper around fast quasi-modp functions, with fall-back to mbedtls_mpi_mod_mpi.
 * See the documentation of struct mbedtls_ecp_group.
 *
 * This function is in the critial loop for mbedtls_ecp_mul, so pay attention to perf.
 */
static int ecp_modp(mbedtls_mpi *N, const mbedtls_ecp_group *grp) {
    int ret;

    if (grp->modp == NULL)
        return (mbedtls_mpi_mod_mpi(N, N, &grp->P));

    /* N->s < 0 is a much faster test, which fails only if N is 0 */
    if ((N->s < 0 && mbedtls_mpi_cmp_int(N, 0) != 0) ||
            mbedtls_mpi_bitlen(N) > 2 * grp->pbits) {
        return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);
    }

    MBEDTLS_MPI_CHK(grp->modp(N));

    /* N->s < 0 is a much faster test, which fails only if N is 0 */
    while (N->s < 0 && mbedtls_mpi_cmp_int(N, 0) != 0)
        MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(N, N, &grp->P));

    while (mbedtls_mpi_cmp_mpi(N, &grp->P) >= 0)
        /* we known P, N and the result are positive */
        MBEDTLS_MPI_CHK(mbedtls_mpi_sub_abs(N, N, &grp->P));

cleanup:
    return (ret);
}

/*
 * Fast mod-p functions expect their argument to be in the 0..p^2 range.
 *
 * In order to guarantee that, we need to ensure that operands of
 * mbedtls_mpi_mul_mpi are in the 0..p range. So, after each operation we will
 * bring the result back to this range.
 *
 * The following macros are shortcuts for doing that.
 */

/*
 * Reduce a mbedtls_mpi mod p in-place, general case, to use after mbedtls_mpi_mul_mpi
 */
#if defined(MBEDTLS_SELF_TEST)
#define INC_MUL_COUNT   mul_count++;
#else
#define INC_MUL_COUNT
#endif

#define MOD_MUL( N )    do { MBEDTLS_MPI_CHK( ecp_modp( &N, grp ) ); INC_MUL_COUNT } \
    while( 0 )

/*
 * Reduce a mbedtls_mpi mod p in-place, to use after mbedtls_mpi_sub_mpi
 * N->s < 0 is a very fast test, which fails only if N is 0
 */
#define MOD_SUB( N )                                \
    while( N.s < 0 && mbedtls_mpi_cmp_int( &N, 0 ) != 0 )   \
        MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &N, &N, &grp->P ) )

/*
 * Reduce a mbedtls_mpi mod p in-place, to use after mbedtls_mpi_add_mpi and mbedtls_mpi_mul_int.
 * We known P, N and the result are positive, so sub_abs is correct, and
 * a bit faster.
 */
#define MOD_ADD( N )                                \
    while( mbedtls_mpi_cmp_mpi( &N, &grp->P ) >= 0 )        \
        MBEDTLS_MPI_CHK( mbedtls_mpi_sub_abs( &N, &N, &grp->P ) )

#if defined(ECP_SHORTWEIERSTRASS)
/*
 * For curves in short Weierstrass form, we do all the internal operations in
 * Jacobian coordinates.
 *
 * For multiplication, we'll use a comb method with coutermeasueres against
 * SPA, hence timing attacks.
 */

/*
 * Normalize jacobian coordinates so that Z == 0 || Z == 1  (GECC 3.2.1)
 * Cost: 1N := 1I + 3M + 1S
 */
static int ecp_normalize_jac(const mbedtls_ecp_group *grp, mbedtls_ecp_point *pt) {
    int ret;
    mbedtls_mpi Zi, ZZi;

    if (mbedtls_mpi_cmp_int(&pt->Z, 0) == 0)
        return (0);

#if defined(MBEDTLS_ECP_NORMALIZE_JAC_ALT)
    if (mbedtls_internal_ecp_grp_capable(grp)) {
        return mbedtls_internal_ecp_normalize_jac(grp, pt);
    }
#endif /* MBEDTLS_ECP_NORMALIZE_JAC_ALT */
    mbedtls_mpi_init(&Zi);
    mbedtls_mpi_init(&ZZi);

    /*
     * X = X / Z^2  mod p
     */
    MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod(&Zi,      &pt->Z,     &grp->P));
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&ZZi,     &Zi,        &Zi));
    MOD_MUL(ZZi);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&pt->X,   &pt->X,     &ZZi));
    MOD_MUL(pt->X);

    /*
     * Y = Y / Z^3  mod p
     */
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&pt->Y,   &pt->Y,     &ZZi));
    MOD_MUL(pt->Y);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&pt->Y,   &pt->Y,     &Zi));
    MOD_MUL(pt->Y);

    /*
     * Z = 1
     */
    MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&pt->Z, 1));

cleanup:

    mbedtls_mpi_free(&Zi);
    mbedtls_mpi_free(&ZZi);

    return (ret);
}

/*
 * Normalize jacobian coordinates of an array of (pointers to) points,
 * using Montgomery's trick to perform only one inversion mod P.
 * (See for example Cohen's "A Course in Computational Algebraic Number
 * Theory", Algorithm 10.3.4.)
 *
 * Warning: fails (returning an error) if one of the points is zero!
 * This should never happen, see choice of w in ecp_mul_comb().
 *
 * Cost: 1N(t) := 1I + (6t - 3)M + 1S
 */
static int ecp_normalize_jac_many(const mbedtls_ecp_group *grp,
                                  mbedtls_ecp_point *T[], size_t t_len) {
    int ret;
    size_t i;
    mbedtls_mpi *c, u, Zi, ZZi;

    if (t_len < 2)
        return (ecp_normalize_jac(grp, *T));

#if defined(MBEDTLS_ECP_NORMALIZE_JAC_MANY_ALT)
    if (mbedtls_internal_ecp_grp_capable(grp)) {
        return mbedtls_internal_ecp_normalize_jac_many(grp, T, t_len);
    }
#endif

    if ((c = mbedtls_calloc(t_len, sizeof(mbedtls_mpi))) == NULL)
        return (MBEDTLS_ERR_ECP_ALLOC_FAILED);

    mbedtls_mpi_init(&u);
    mbedtls_mpi_init(&Zi);
    mbedtls_mpi_init(&ZZi);

    /*
     * c[i] = Z_0 * ... * Z_i
     */
    MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&c[0], &T[0]->Z));
    for (i = 1; i < t_len; i++) {
        MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&c[i], &c[i - 1], &T[i]->Z));
        MOD_MUL(c[i]);
    }

    /*
     * u = 1 / (Z_0 * ... * Z_n) mod P
     */
    MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod(&u, &c[t_len - 1], &grp->P));

    for (i = t_len - 1; ; i--) {
        /*
         * Zi = 1 / Z_i mod p
         * u = 1 / (Z_0 * ... * Z_i) mod P
         */
        if (i == 0) {
            MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&Zi, &u));
        } else {
            MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&Zi, &u, &c[i - 1]));
            MOD_MUL(Zi);
            MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&u,  &u, &T[i]->Z));
            MOD_MUL(u);
        }

        /*
         * proceed as in normalize()
         */
        MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&ZZi,     &Zi,      &Zi));
        MOD_MUL(ZZi);
        MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T[i]->X, &T[i]->X, &ZZi));
        MOD_MUL(T[i]->X);
        MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T[i]->Y, &T[i]->Y, &ZZi));
        MOD_MUL(T[i]->Y);
        MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T[i]->Y, &T[i]->Y, &Zi));
        MOD_MUL(T[i]->Y);

        /*
         * Post-precessing: reclaim some memory by shrinking coordinates
         * - not storing Z (always 1)
         * - shrinking other coordinates, but still keeping the same number of
         *   limbs as P, as otherwise it will too likely be regrown too fast.
         */
        MBEDTLS_MPI_CHK(mbedtls_mpi_shrink(&T[i]->X, grp->P.n));
        MBEDTLS_MPI_CHK(mbedtls_mpi_shrink(&T[i]->Y, grp->P.n));
        mbedtls_mpi_free(&T[i]->Z);

        if (i == 0)
            break;
    }

cleanup:

    mbedtls_mpi_free(&u);
    mbedtls_mpi_free(&Zi);
    mbedtls_mpi_free(&ZZi);
    for (i = 0; i < t_len; i++)
        mbedtls_mpi_free(&c[i]);
    mbedtls_free(c);

    return (ret);
}

/*
 * Conditional point inversion: Q -> -Q = (Q.X, -Q.Y, Q.Z) without leak.
 * "inv" must be 0 (don't invert) or 1 (invert) or the result will be invalid
 */
static int ecp_safe_invert_jac(const mbedtls_ecp_group *grp,
                               mbedtls_ecp_point *Q,
                               unsigned char inv) {
    int ret;
    unsigned char nonzero;
    mbedtls_mpi mQY;

    mbedtls_mpi_init(&mQY);

    /* Use the fact that -Q.Y mod P = P - Q.Y unless Q.Y == 0 */
    MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&mQY, &grp->P, &Q->Y));
    nonzero = mbedtls_mpi_cmp_int(&Q->Y, 0) != 0;
    MBEDTLS_MPI_CHK(mbedtls_mpi_safe_cond_assign(&Q->Y, &mQY, inv & nonzero));

cleanup:
    mbedtls_mpi_free(&mQY);

    return (ret);
}

/*
 * Point doubling R = 2 P, Jacobian coordinates
 *
 * Based on http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html#doubling-dbl-1998-cmo-2 .
 *
 * We follow the variable naming fairly closely. The formula variations that trade a MUL for a SQR
 * (plus a few ADDs) aren't useful as our bignum implementation doesn't distinguish squaring.
 *
 * Standard optimizations are applied when curve parameter A is one of { 0, -3 }.
 *
 * Cost: 1D := 3M + 4S          (A ==  0)
 *             4M + 4S          (A == -3)
 *             3M + 6S + 1a     otherwise
 */
static int ecp_double_jac(const mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
                          const mbedtls_ecp_point *P) {
    int ret;
    mbedtls_mpi M, S, T, U;

#if defined(MBEDTLS_SELF_TEST)
    dbl_count++;
#endif

#if defined(MBEDTLS_ECP_DOUBLE_JAC_ALT)
    if (mbedtls_internal_ecp_grp_capable(grp)) {
        return mbedtls_internal_ecp_double_jac(grp, R, P);
    }
#endif /* MBEDTLS_ECP_DOUBLE_JAC_ALT */

    mbedtls_mpi_init(&M);
    mbedtls_mpi_init(&S);
    mbedtls_mpi_init(&T);
    mbedtls_mpi_init(&U);

    /* Special case for A = -3 */
    if (grp->A.p == NULL) {
        /* M = 3(X + Z^2)(X - Z^2) */
        MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&S,  &P->Z,  &P->Z));
        MOD_MUL(S);
        MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(&T,  &P->X,  &S));
        MOD_ADD(T);
        MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&U,  &P->X,  &S));
        MOD_SUB(U);
        MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&S,  &T,     &U));
        MOD_MUL(S);
        MBEDTLS_MPI_CHK(mbedtls_mpi_mul_int(&M,  &S,     3));
        MOD_ADD(M);
    } else {
        /* M = 3.X^2 */
        MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&S,  &P->X,  &P->X));
        MOD_MUL(S);
        MBEDTLS_MPI_CHK(mbedtls_mpi_mul_int(&M,  &S,     3));
        MOD_ADD(M);

        /* Optimize away for "koblitz" curves with A = 0 */
        if (mbedtls_mpi_cmp_int(&grp->A, 0) != 0) {
            /* M += A.Z^4 */
            MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&S,  &P->Z,  &P->Z));
            MOD_MUL(S);
            MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T,  &S,     &S));
            MOD_MUL(T);
            MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&S,  &T,     &grp->A));
            MOD_MUL(S);
            MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(&M,  &M,     &S));
            MOD_ADD(M);
        }
    }

    /* S = 4.X.Y^2 */
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T,  &P->Y,  &P->Y));
    MOD_MUL(T);
    MBEDTLS_MPI_CHK(mbedtls_mpi_shift_l(&T,  1));
    MOD_ADD(T);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&S,  &P->X,  &T));
    MOD_MUL(S);
    MBEDTLS_MPI_CHK(mbedtls_mpi_shift_l(&S,  1));
    MOD_ADD(S);

    /* U = 8.Y^4 */
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&U,  &T,     &T));
    MOD_MUL(U);
    MBEDTLS_MPI_CHK(mbedtls_mpi_shift_l(&U,  1));
    MOD_ADD(U);

    /* T = M^2 - 2.S */
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T,  &M,     &M));
    MOD_MUL(T);
    MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&T,  &T,     &S));
    MOD_SUB(T);
    MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&T,  &T,     &S));
    MOD_SUB(T);

    /* S = M(S - T) - U */
    MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&S,  &S,     &T));
    MOD_SUB(S);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&S,  &S,     &M));
    MOD_MUL(S);
    MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&S,  &S,     &U));
    MOD_SUB(S);

    /* U = 2.Y.Z */
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&U,  &P->Y,  &P->Z));
    MOD_MUL(U);
    MBEDTLS_MPI_CHK(mbedtls_mpi_shift_l(&U,  1));
    MOD_ADD(U);

    MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&R->X, &T));
    MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&R->Y, &S));
    MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&R->Z, &U));

cleanup:
    mbedtls_mpi_free(&M);
    mbedtls_mpi_free(&S);
    mbedtls_mpi_free(&T);
    mbedtls_mpi_free(&U);

    return (ret);
}

/*
 * Addition: R = P + Q, mixed affine-Jacobian coordinates (GECC 3.22)
 *
 * The coordinates of Q must be normalized (= affine),
 * but those of P don't need to. R is not normalized.
 *
 * Special cases: (1) P or Q is zero, (2) R is zero, (3) P == Q.
 * None of these cases can happen as intermediate step in ecp_mul_comb():
 * - at each step, P, Q and R are multiples of the base point, the factor
 *   being less than its order, so none of them is zero;
 * - Q is an odd multiple of the base point, P an even multiple,
 *   due to the choice of precomputed points in the modified comb method.
 * So branches for these cases do not leak secret information.
 *
 * We accept Q->Z being unset (saving memory in tables) as meaning 1.
 *
 * Cost: 1A := 8M + 3S
 */
static int ecp_add_mixed(const mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
                         const mbedtls_ecp_point *P, const mbedtls_ecp_point *Q) {
    int ret;
    mbedtls_mpi T1, T2, T3, T4, X, Y, Z;

#if defined(MBEDTLS_SELF_TEST)
    add_count++;
#endif

#if defined(MBEDTLS_ECP_ADD_MIXED_ALT)
    if (mbedtls_internal_ecp_grp_capable(grp)) {
        return mbedtls_internal_ecp_add_mixed(grp, R, P, Q);
    }
#endif /* MBEDTLS_ECP_ADD_MIXED_ALT */

    /*
     * Trivial cases: P == 0 or Q == 0 (case 1)
     */
    if (mbedtls_mpi_cmp_int(&P->Z, 0) == 0)
        return (mbedtls_ecp_copy(R, Q));

    if (Q->Z.p != NULL && mbedtls_mpi_cmp_int(&Q->Z, 0) == 0)
        return (mbedtls_ecp_copy(R, P));

    /*
     * Make sure Q coordinates are normalized
     */
    if (Q->Z.p != NULL && mbedtls_mpi_cmp_int(&Q->Z, 1) != 0)
        return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);

    mbedtls_mpi_init(&T1);
    mbedtls_mpi_init(&T2);
    mbedtls_mpi_init(&T3);
    mbedtls_mpi_init(&T4);
    mbedtls_mpi_init(&X);
    mbedtls_mpi_init(&Y);
    mbedtls_mpi_init(&Z);

    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T1,  &P->Z,  &P->Z));
    MOD_MUL(T1);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T2,  &T1,    &P->Z));
    MOD_MUL(T2);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T1,  &T1,    &Q->X));
    MOD_MUL(T1);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T2,  &T2,    &Q->Y));
    MOD_MUL(T2);
    MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&T1,  &T1,    &P->X));
    MOD_SUB(T1);
    MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&T2,  &T2,    &P->Y));
    MOD_SUB(T2);

    /* Special cases (2) and (3) */
    if (mbedtls_mpi_cmp_int(&T1, 0) == 0) {
        if (mbedtls_mpi_cmp_int(&T2, 0) == 0) {
            ret = ecp_double_jac(grp, R, P);
            goto cleanup;
        } else {
            ret = mbedtls_ecp_set_zero(R);
            goto cleanup;
        }
    }

    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&Z,   &P->Z,  &T1));
    MOD_MUL(Z);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T3,  &T1,    &T1));
    MOD_MUL(T3);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T4,  &T3,    &T1));
    MOD_MUL(T4);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T3,  &T3,    &P->X));
    MOD_MUL(T3);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_int(&T1,  &T3,    2));
    MOD_ADD(T1);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&X,   &T2,    &T2));
    MOD_MUL(X);
    MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&X,   &X,     &T1));
    MOD_SUB(X);
    MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&X,   &X,     &T4));
    MOD_SUB(X);
    MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&T3,  &T3,    &X));
    MOD_SUB(T3);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T3,  &T3,    &T2));
    MOD_MUL(T3);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T4,  &T4,    &P->Y));
    MOD_MUL(T4);
    MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&Y,   &T3,    &T4));
    MOD_SUB(Y);

    MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&R->X, &X));
    MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&R->Y, &Y));
    MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&R->Z, &Z));

cleanup:

    mbedtls_mpi_free(&T1);
    mbedtls_mpi_free(&T2);
    mbedtls_mpi_free(&T3);
    mbedtls_mpi_free(&T4);
    mbedtls_mpi_free(&X);
    mbedtls_mpi_free(&Y);
    mbedtls_mpi_free(&Z);

    return (ret);
}

/*
 * Randomize jacobian coordinates:
 * (X, Y, Z) -> (l^2 X, l^3 Y, l Z) for random l
 * This is sort of the reverse operation of ecp_normalize_jac().
 *
 * This countermeasure was first suggested in [2].
 */
static int ecp_randomize_jac(const mbedtls_ecp_group *grp, mbedtls_ecp_point *pt,
                             int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) {
    int ret;
    mbedtls_mpi l, ll;
    size_t p_size;
    int count = 0;

#if defined(MBEDTLS_ECP_RANDOMIZE_JAC_ALT)
    if (mbedtls_internal_ecp_grp_capable(grp)) {
        return mbedtls_internal_ecp_randomize_jac(grp, pt, f_rng, p_rng);
    }
#endif /* MBEDTLS_ECP_RANDOMIZE_JAC_ALT */

    p_size = (grp->pbits + 7) / 8;
    mbedtls_mpi_init(&l);
    mbedtls_mpi_init(&ll);

    /* Generate l such that 1 < l < p */
    do {
        MBEDTLS_MPI_CHK(mbedtls_mpi_fill_random(&l, p_size, f_rng, p_rng));

        while (mbedtls_mpi_cmp_mpi(&l, &grp->P) >= 0)
            MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&l, 1));

        if (count++ > 10)
            return (MBEDTLS_ERR_ECP_RANDOM_FAILED);
    } while (mbedtls_mpi_cmp_int(&l, 1) <= 0);

    /* Z = l * Z */
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&pt->Z,   &pt->Z,     &l));
    MOD_MUL(pt->Z);

    /* X = l^2 * X */
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&ll,      &l,         &l));
    MOD_MUL(ll);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&pt->X,   &pt->X,     &ll));
    MOD_MUL(pt->X);

    /* Y = l^3 * Y */
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&ll,      &ll,        &l));
    MOD_MUL(ll);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&pt->Y,   &pt->Y,     &ll));
    MOD_MUL(pt->Y);

cleanup:
    mbedtls_mpi_free(&l);
    mbedtls_mpi_free(&ll);

    return (ret);
}

/*
 * Check and define parameters used by the comb method (see below for details)
 */
#if MBEDTLS_ECP_WINDOW_SIZE < 2 || MBEDTLS_ECP_WINDOW_SIZE > 7
#error "MBEDTLS_ECP_WINDOW_SIZE out of bounds"
#endif

/* d = ceil( n / w ) */
#define COMB_MAX_D      ( MBEDTLS_ECP_MAX_BITS + 1 ) / 2

/* number of precomputed points */
#define COMB_MAX_PRE    ( 1 << ( MBEDTLS_ECP_WINDOW_SIZE - 1 ) )

/*
 * Compute the representation of m that will be used with our comb method.
 *
 * The basic comb method is described in GECC 3.44 for example. We use a
 * modified version that provides resistance to SPA by avoiding zero
 * digits in the representation as in [3]. We modify the method further by
 * requiring that all K_i be odd, which has the small cost that our
 * representation uses one more K_i, due to carries.
 *
 * Also, for the sake of compactness, only the seven low-order bits of x[i]
 * are used to represent K_i, and the msb of x[i] encodes the the sign (s_i in
 * the paper): it is set if and only if if s_i == -1;
 *
 * Calling conventions:
 * - x is an array of size d + 1
 * - w is the size, ie number of teeth, of the comb, and must be between
 *   2 and 7 (in practice, between 2 and MBEDTLS_ECP_WINDOW_SIZE)
 * - m is the MPI, expected to be odd and such that bitlength(m) <= w * d
 *   (the result will be incorrect if these assumptions are not satisfied)
 */
static void ecp_comb_fixed(unsigned char x[], size_t d,
                           unsigned char w, const mbedtls_mpi *m) {
    size_t i, j;
    unsigned char c, cc, adjust;

    memset(x, 0, d + 1);

    /* First get the classical comb values (except for x_d = 0) */
    for (i = 0; i < d; i++)
        for (j = 0; j < w; j++)
            x[i] |= mbedtls_mpi_get_bit(m, i + d * j) << j;

    /* Now make sure x_1 .. x_d are odd */
    c = 0;
    for (i = 1; i <= d; i++) {
        /* Add carry and update it */
        cc   = x[i] & c;
        x[i] = x[i] ^ c;
        c = cc;

        /* Adjust if needed, avoiding branches */
        adjust = 1 - (x[i] & 0x01);
        c   |= x[i] & (x[i - 1] * adjust);
        x[i] = x[i] ^ (x[i - 1] * adjust);
        x[i - 1] |= adjust << 7;
    }
}

/*
 * Precompute points for the comb method
 *
 * If i = i_{w-1} ... i_1 is the binary representation of i, then
 * T[i] = i_{w-1} 2^{(w-1)d} P + ... + i_1 2^d P + P
 *
 * T must be able to hold 2^{w - 1} elements
 *
 * Cost: d(w-1) D + (2^{w-1} - 1) A + 1 N(w-1) + 1 N(2^{w-1} - 1)
 */
static int ecp_precompute_comb(const mbedtls_ecp_group *grp,
                               mbedtls_ecp_point T[], const mbedtls_ecp_point *P,
                               unsigned char w, size_t d) {
    int ret;
    unsigned char i, k;
    size_t j;
    mbedtls_ecp_point *cur, *TT[COMB_MAX_PRE - 1];

    /*
     * Set T[0] = P and
     * T[2^{l-1}] = 2^{dl} P for l = 1 .. w-1 (this is not the final value)
     */
    MBEDTLS_MPI_CHK(mbedtls_ecp_copy(&T[0], P));

    k = 0;
    for (i = 1; i < (1U << (w - 1)); i <<= 1) {
        cur = T + i;
        MBEDTLS_MPI_CHK(mbedtls_ecp_copy(cur, T + (i >> 1)));
        for (j = 0; j < d; j++)
            MBEDTLS_MPI_CHK(ecp_double_jac(grp, cur, cur));

        TT[k++] = cur;
    }

    MBEDTLS_MPI_CHK(ecp_normalize_jac_many(grp, TT, k));

    /*
     * Compute the remaining ones using the minimal number of additions
     * Be careful to update T[2^l] only after using it!
     */
    k = 0;
    for (i = 1; i < (1U << (w - 1)); i <<= 1) {
        j = i;
        while (j--) {
            MBEDTLS_MPI_CHK(ecp_add_mixed(grp, &T[i + j], &T[j], &T[i]));
            TT[k++] = &T[i + j];
        }
    }

    MBEDTLS_MPI_CHK(ecp_normalize_jac_many(grp, TT, k));

cleanup:

    return (ret);
}

/*
 * Select precomputed point: R = sign(i) * T[ abs(i) / 2 ]
 */
static int ecp_select_comb(const mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
                           const mbedtls_ecp_point T[], unsigned char t_len,
                           unsigned char i) {
    int ret;
    unsigned char ii, j;

    /* Ignore the "sign" bit and scale down */
    ii = (i & 0x7Fu) >> 1;

    /* Read the whole table to thwart cache-based timing attacks */
    for (j = 0; j < t_len; j++) {
        MBEDTLS_MPI_CHK(mbedtls_mpi_safe_cond_assign(&R->X, &T[j].X, j == ii));
        MBEDTLS_MPI_CHK(mbedtls_mpi_safe_cond_assign(&R->Y, &T[j].Y, j == ii));
    }

    /* Safely invert result if i is "negative" */
    MBEDTLS_MPI_CHK(ecp_safe_invert_jac(grp, R, i >> 7));

cleanup:
    return (ret);
}

/*
 * Core multiplication algorithm for the (modified) comb method.
 * This part is actually common with the basic comb method (GECC 3.44)
 *
 * Cost: d A + d D + 1 R
 */
static int ecp_mul_comb_core(const mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
                             const mbedtls_ecp_point T[], unsigned char t_len,
                             const unsigned char x[], size_t d,
                             int (*f_rng)(void *, unsigned char *, size_t),
                             void *p_rng) {
    int ret;
    mbedtls_ecp_point Txi;
    size_t i;

    mbedtls_ecp_point_init(&Txi);

    /* Start with a non-zero point and randomize its coordinates */
    i = d;
    MBEDTLS_MPI_CHK(ecp_select_comb(grp, R, T, t_len, x[i]));
    MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&R->Z, 1));
    if (f_rng != 0)
        MBEDTLS_MPI_CHK(ecp_randomize_jac(grp, R, f_rng, p_rng));

    while (i-- != 0) {
        MBEDTLS_MPI_CHK(ecp_double_jac(grp, R, R));
        MBEDTLS_MPI_CHK(ecp_select_comb(grp, &Txi, T, t_len, x[i]));
        MBEDTLS_MPI_CHK(ecp_add_mixed(grp, R, R, &Txi));
    }

cleanup:

    mbedtls_ecp_point_free(&Txi);

    return (ret);
}

/*
 * Multiplication using the comb method,
 * for curves in short Weierstrass form
 */
static int ecp_mul_comb(mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
                        const mbedtls_mpi *m, const mbedtls_ecp_point *P,
                        int (*f_rng)(void *, unsigned char *, size_t),
                        void *p_rng) {
    int ret;
    unsigned char w, m_is_odd, p_eq_g, pre_len, i;
    size_t d;
    unsigned char k[COMB_MAX_D + 1];
    mbedtls_ecp_point *T;
    mbedtls_mpi M, mm;

    mbedtls_mpi_init(&M);
    mbedtls_mpi_init(&mm);

    /* we need N to be odd to trnaform m in an odd number, check now */
    if (mbedtls_mpi_get_bit(&grp->N, 0) != 1)
        return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);

    /*
     * Minimize the number of multiplications, that is minimize
     * 10 * d * w + 18 * 2^(w-1) + 11 * d + 7 * w, with d = ceil( nbits / w )
     * (see costs of the various parts, with 1S = 1M)
     */
    w = grp->nbits >= 384 ? 5 : 4;

    /*
     * If P == G, pre-compute a bit more, since this may be re-used later.
     * Just adding one avoids upping the cost of the first mul too much,
     * and the memory cost too.
     */
#if MBEDTLS_ECP_FIXED_POINT_OPTIM == 1
    p_eq_g = (mbedtls_mpi_cmp_mpi(&P->Y, &grp->G.Y) == 0 &&
              mbedtls_mpi_cmp_mpi(&P->X, &grp->G.X) == 0);
    if (p_eq_g)
        w++;
#else
    p_eq_g = 0;
#endif

    /*
     * Make sure w is within bounds.
     * (The last test is useful only for very small curves in the test suite.)
     */
    if (w > MBEDTLS_ECP_WINDOW_SIZE)
        w = MBEDTLS_ECP_WINDOW_SIZE;
    if (w >= grp->nbits)
        w = 2;

    /* Other sizes that depend on w */
    pre_len = 1U << (w - 1);
    d = (grp->nbits + w - 1) / w;

    /*
     * Prepare precomputed points: if P == G we want to
     * use grp->T if already initialized, or initialize it.
     */
    T = p_eq_g ? grp->T : NULL;

    if (T == NULL) {
        T = mbedtls_calloc(pre_len, sizeof(mbedtls_ecp_point));
        if (T == NULL) {
            ret = MBEDTLS_ERR_ECP_ALLOC_FAILED;
            goto cleanup;
        }

        MBEDTLS_MPI_CHK(ecp_precompute_comb(grp, T, P, w, d));

        if (p_eq_g) {
            grp->T = T;
            grp->T_size = pre_len;
        }
    }

    /*
     * Make sure M is odd (M = m or M = N - m, since N is odd)
     * using the fact that m * P = - (N - m) * P
     */
    m_is_odd = (mbedtls_mpi_get_bit(m, 0) == 1);
    MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&M, m));
    MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&mm, &grp->N, m));
    MBEDTLS_MPI_CHK(mbedtls_mpi_safe_cond_assign(&M, &mm, ! m_is_odd));

    /*
     * Go for comb multiplication, R = M * P
     */
    ecp_comb_fixed(k, d, w, &M);
    MBEDTLS_MPI_CHK(ecp_mul_comb_core(grp, R, T, pre_len, k, d, f_rng, p_rng));

    /*
     * Now get m * P from M * P and normalize it
     */
    MBEDTLS_MPI_CHK(ecp_safe_invert_jac(grp, R, ! m_is_odd));
    MBEDTLS_MPI_CHK(ecp_normalize_jac(grp, R));

cleanup:

    /* There are two cases where T is not stored in grp:
     * - P != G
     * - An intermediate operation failed before setting grp->T
     * In either case, T must be freed.
     */
    if (T != NULL && T != grp->T) {
        for (i = 0; i < pre_len; i++)
            mbedtls_ecp_point_free(&T[i]);
        mbedtls_free(T);
    }

    mbedtls_mpi_free(&M);
    mbedtls_mpi_free(&mm);

    if (ret != 0)
        mbedtls_ecp_point_free(R);

    return (ret);
}

#endif /* ECP_SHORTWEIERSTRASS */

#if defined(ECP_MONTGOMERY)
/*
 * For Montgomery curves, we do all the internal arithmetic in projective
 * coordinates. Import/export of points uses only the x coordinates, which is
 * internaly represented as X / Z.
 *
 * For scalar multiplication, we'll use a Montgomery ladder.
 */

/*
 * Normalize Montgomery x/z coordinates: X = X/Z, Z = 1
 * Cost: 1M + 1I
 */
static int ecp_normalize_mxz(const mbedtls_ecp_group *grp, mbedtls_ecp_point *P) {
    int ret;

#if defined(MBEDTLS_ECP_NORMALIZE_MXZ_ALT)
    if (mbedtls_internal_ecp_grp_capable(grp)) {
        return mbedtls_internal_ecp_normalize_mxz(grp, P);
    }
#endif /* MBEDTLS_ECP_NORMALIZE_MXZ_ALT */

    MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod(&P->Z, &P->Z, &grp->P));
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&P->X, &P->X, &P->Z));
    MOD_MUL(P->X);
    MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&P->Z, 1));

cleanup:
    return (ret);
}

/*
 * Randomize projective x/z coordinates:
 * (X, Z) -> (l X, l Z) for random l
 * This is sort of the reverse operation of ecp_normalize_mxz().
 *
 * This countermeasure was first suggested in [2].
 * Cost: 2M
 */
static int ecp_randomize_mxz(const mbedtls_ecp_group *grp, mbedtls_ecp_point *P,
                             int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) {
    int ret;
    mbedtls_mpi l;
    size_t p_size;
    int count = 0;

#if defined(MBEDTLS_ECP_RANDOMIZE_MXZ_ALT)
    if (mbedtls_internal_ecp_grp_capable(grp)) {
        return mbedtls_internal_ecp_randomize_mxz(grp, P, f_rng, p_rng);
    }
#endif /* MBEDTLS_ECP_RANDOMIZE_MXZ_ALT */

    p_size = (grp->pbits + 7) / 8;
    mbedtls_mpi_init(&l);

    /* Generate l such that 1 < l < p */
    do {
        MBEDTLS_MPI_CHK(mbedtls_mpi_fill_random(&l, p_size, f_rng, p_rng));

        while (mbedtls_mpi_cmp_mpi(&l, &grp->P) >= 0)
            MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&l, 1));

        if (count++ > 10)
            return (MBEDTLS_ERR_ECP_RANDOM_FAILED);
    } while (mbedtls_mpi_cmp_int(&l, 1) <= 0);

    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&P->X, &P->X, &l));
    MOD_MUL(P->X);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&P->Z, &P->Z, &l));
    MOD_MUL(P->Z);

cleanup:
    mbedtls_mpi_free(&l);

    return (ret);
}

/*
 * Double-and-add: R = 2P, S = P + Q, with d = X(P - Q),
 * for Montgomery curves in x/z coordinates.
 *
 * http://www.hyperelliptic.org/EFD/g1p/auto-code/montgom/xz/ladder/mladd-1987-m.op3
 * with
 * d =  X1
 * P = (X2, Z2)
 * Q = (X3, Z3)
 * R = (X4, Z4)
 * S = (X5, Z5)
 * and eliminating temporary variables tO, ..., t4.
 *
 * Cost: 5M + 4S
 */
static int ecp_double_add_mxz(const mbedtls_ecp_group *grp,
                              mbedtls_ecp_point *R, mbedtls_ecp_point *S,
                              const mbedtls_ecp_point *P, const mbedtls_ecp_point *Q,
                              const mbedtls_mpi *d) {
    int ret;
    mbedtls_mpi A, AA, B, BB, E, C, D, DA, CB;

#if defined(MBEDTLS_ECP_DOUBLE_ADD_MXZ_ALT)
    if (mbedtls_internal_ecp_grp_capable(grp)) {
        return mbedtls_internal_ecp_double_add_mxz(grp, R, S, P, Q, d);
    }
#endif /* MBEDTLS_ECP_DOUBLE_ADD_MXZ_ALT */

    mbedtls_mpi_init(&A);
    mbedtls_mpi_init(&AA);
    mbedtls_mpi_init(&B);
    mbedtls_mpi_init(&BB);
    mbedtls_mpi_init(&E);
    mbedtls_mpi_init(&C);
    mbedtls_mpi_init(&D);
    mbedtls_mpi_init(&DA);
    mbedtls_mpi_init(&CB);

    MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(&A,    &P->X,   &P->Z));
    MOD_ADD(A);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&AA,   &A,      &A));
    MOD_MUL(AA);
    MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&B,    &P->X,   &P->Z));
    MOD_SUB(B);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&BB,   &B,      &B));
    MOD_MUL(BB);
    MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&E,    &AA,     &BB));
    MOD_SUB(E);
    MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(&C,    &Q->X,   &Q->Z));
    MOD_ADD(C);
    MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&D,    &Q->X,   &Q->Z));
    MOD_SUB(D);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&DA,   &D,      &A));
    MOD_MUL(DA);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&CB,   &C,      &B));
    MOD_MUL(CB);
    MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(&S->X, &DA,     &CB));
    MOD_MUL(S->X);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&S->X, &S->X,   &S->X));
    MOD_MUL(S->X);
    MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&S->Z, &DA,     &CB));
    MOD_SUB(S->Z);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&S->Z, &S->Z,   &S->Z));
    MOD_MUL(S->Z);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&S->Z, d,       &S->Z));
    MOD_MUL(S->Z);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&R->X, &AA,     &BB));
    MOD_MUL(R->X);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&R->Z, &grp->A, &E));
    MOD_MUL(R->Z);
    MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(&R->Z, &BB,     &R->Z));
    MOD_ADD(R->Z);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&R->Z, &E,      &R->Z));
    MOD_MUL(R->Z);

cleanup:
    mbedtls_mpi_free(&A);
    mbedtls_mpi_free(&AA);
    mbedtls_mpi_free(&B);
    mbedtls_mpi_free(&BB);
    mbedtls_mpi_free(&E);
    mbedtls_mpi_free(&C);
    mbedtls_mpi_free(&D);
    mbedtls_mpi_free(&DA);
    mbedtls_mpi_free(&CB);

    return (ret);
}

/*
 * Multiplication with Montgomery ladder in x/z coordinates,
 * for curves in Montgomery form
 */
static int ecp_mul_mxz(mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
                       const mbedtls_mpi *m, const mbedtls_ecp_point *P,
                       int (*f_rng)(void *, unsigned char *, size_t),
                       void *p_rng) {
    int ret;
    size_t i;
    unsigned char b;
    mbedtls_ecp_point RP;
    mbedtls_mpi PX;

    mbedtls_ecp_point_init(&RP);
    mbedtls_mpi_init(&PX);

    /* Save PX and read from P before writing to R, in case P == R */
    MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&PX, &P->X));
    MBEDTLS_MPI_CHK(mbedtls_ecp_copy(&RP, P));

    /* Set R to zero in modified x/z coordinates */
    MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&R->X, 1));
    MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&R->Z, 0));
    mbedtls_mpi_free(&R->Y);

    /* RP.X might be sligtly larger than P, so reduce it */
    MOD_ADD(RP.X);

    /* Randomize coordinates of the starting point */
    if (f_rng != NULL)
        MBEDTLS_MPI_CHK(ecp_randomize_mxz(grp, &RP, f_rng, p_rng));

    /* Loop invariant: R = result so far, RP = R + P */
    i = mbedtls_mpi_bitlen(m);   /* one past the (zero-based) most significant bit */
    while (i-- > 0) {
        b = mbedtls_mpi_get_bit(m, i);
        /*
         *  if (b) R = 2R + P else R = 2R,
         * which is:
         *  if (b) double_add( RP, R, RP, R )
         *  else   double_add( R, RP, R, RP )
         * but using safe conditional swaps to avoid leaks
         */
        MBEDTLS_MPI_CHK(mbedtls_mpi_safe_cond_swap(&R->X, &RP.X, b));
        MBEDTLS_MPI_CHK(mbedtls_mpi_safe_cond_swap(&R->Z, &RP.Z, b));
        MBEDTLS_MPI_CHK(ecp_double_add_mxz(grp, R, &RP, R, &RP, &PX));
        MBEDTLS_MPI_CHK(mbedtls_mpi_safe_cond_swap(&R->X, &RP.X, b));
        MBEDTLS_MPI_CHK(mbedtls_mpi_safe_cond_swap(&R->Z, &RP.Z, b));
    }

    MBEDTLS_MPI_CHK(ecp_normalize_mxz(grp, R));

cleanup:
    mbedtls_ecp_point_free(&RP);
    mbedtls_mpi_free(&PX);

    return (ret);
}

#endif /* ECP_MONTGOMERY */

/*
 * Multiplication R = m * P
 */
int mbedtls_ecp_mul(mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
                    const mbedtls_mpi *m, const mbedtls_ecp_point *P,
                    int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) {
    int ret = MBEDTLS_ERR_ECP_BAD_INPUT_DATA;
#if defined(MBEDTLS_ECP_INTERNAL_ALT)
    char is_grp_capable = 0;
#endif

    /* Common sanity checks */
    if (mbedtls_mpi_cmp_int(&P->Z, 1) != 0)
        return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);

    if ((ret = mbedtls_ecp_check_privkey(grp, m)) != 0 ||
            (ret = mbedtls_ecp_check_pubkey(grp, P)) != 0)
        return (ret);

#if defined(MBEDTLS_ECP_INTERNAL_ALT)
    if (is_grp_capable = mbedtls_internal_ecp_grp_capable(grp)) {
        MBEDTLS_MPI_CHK(mbedtls_internal_ecp_init(grp));
    }

#endif /* MBEDTLS_ECP_INTERNAL_ALT */
#if defined(ECP_MONTGOMERY)
    if (ecp_get_type(grp) == ECP_TYPE_MONTGOMERY)
        ret = ecp_mul_mxz(grp, R, m, P, f_rng, p_rng);

#endif
#if defined(ECP_SHORTWEIERSTRASS)
    if (ecp_get_type(grp) == ECP_TYPE_SHORT_WEIERSTRASS)
        ret = ecp_mul_comb(grp, R, m, P, f_rng, p_rng);

#endif
#if defined(MBEDTLS_ECP_INTERNAL_ALT)
cleanup:

    if (is_grp_capable) {
        mbedtls_internal_ecp_free(grp);
    }

#endif /* MBEDTLS_ECP_INTERNAL_ALT */
    return (ret);
}

#if defined(ECP_SHORTWEIERSTRASS)
/*
 * Check that an affine point is valid as a public key,
 * short weierstrass curves (SEC1 3.2.3.1)
 */
static int ecp_check_pubkey_sw(const mbedtls_ecp_group *grp, const mbedtls_ecp_point *pt) {
    int ret;
    mbedtls_mpi YY, RHS;

    /* pt coordinates must be normalized for our checks */
    if (mbedtls_mpi_cmp_int(&pt->X, 0) < 0 ||
            mbedtls_mpi_cmp_int(&pt->Y, 0) < 0 ||
            mbedtls_mpi_cmp_mpi(&pt->X, &grp->P) >= 0 ||
            mbedtls_mpi_cmp_mpi(&pt->Y, &grp->P) >= 0)
        return (MBEDTLS_ERR_ECP_INVALID_KEY);

    mbedtls_mpi_init(&YY);
    mbedtls_mpi_init(&RHS);

    /*
     * YY = Y^2
     * RHS = X (X^2 + A) + B = X^3 + A X + B
     */
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&YY,  &pt->Y,   &pt->Y));
    MOD_MUL(YY);
    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&RHS, &pt->X,   &pt->X));
    MOD_MUL(RHS);

    /* Special case for A = -3 */
    if (grp->A.p == NULL) {
        MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&RHS, &RHS, 3));
        MOD_SUB(RHS);
    } else {
        MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(&RHS, &RHS, &grp->A));
        MOD_ADD(RHS);
    }

    MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&RHS, &RHS,     &pt->X));
    MOD_MUL(RHS);
    MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(&RHS, &RHS,     &grp->B));
    MOD_ADD(RHS);

    if (mbedtls_mpi_cmp_mpi(&YY, &RHS) != 0)
        ret = MBEDTLS_ERR_ECP_INVALID_KEY;

cleanup:

    mbedtls_mpi_free(&YY);
    mbedtls_mpi_free(&RHS);

    return (ret);
}
#endif /* ECP_SHORTWEIERSTRASS */

/*
 * R = m * P with shortcuts for m == 1 and m == -1
 * NOT constant-time - ONLY for short Weierstrass!
 */
static int mbedtls_ecp_mul_shortcuts(mbedtls_ecp_group *grp,
                                     mbedtls_ecp_point *R,
                                     const mbedtls_mpi *m,
                                     const mbedtls_ecp_point *P) {
    int ret;

    if (mbedtls_mpi_cmp_int(m, 1) == 0) {
        MBEDTLS_MPI_CHK(mbedtls_ecp_copy(R, P));
    } else if (mbedtls_mpi_cmp_int(m, -1) == 0) {
        MBEDTLS_MPI_CHK(mbedtls_ecp_copy(R, P));
        if (mbedtls_mpi_cmp_int(&R->Y, 0) != 0)
            MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&R->Y, &grp->P, &R->Y));
    } else {
        MBEDTLS_MPI_CHK(mbedtls_ecp_mul(grp, R, m, P, NULL, NULL));
    }

cleanup:
    return (ret);
}

/*
 * Linear combination
 * NOT constant-time
 */
int mbedtls_ecp_muladd(mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
                       const mbedtls_mpi *m, const mbedtls_ecp_point *P,
                       const mbedtls_mpi *n, const mbedtls_ecp_point *Q) {
    int ret;
    mbedtls_ecp_point mP;
#if defined(MBEDTLS_ECP_INTERNAL_ALT)
    char is_grp_capable = 0;
#endif

    if (ecp_get_type(grp) != ECP_TYPE_SHORT_WEIERSTRASS)
        return (MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE);

    mbedtls_ecp_point_init(&mP);

    MBEDTLS_MPI_CHK(mbedtls_ecp_mul_shortcuts(grp, &mP, m, P));
    MBEDTLS_MPI_CHK(mbedtls_ecp_mul_shortcuts(grp, R,   n, Q));

#if defined(MBEDTLS_ECP_INTERNAL_ALT)
    if (is_grp_capable = mbedtls_internal_ecp_grp_capable(grp)) {
        MBEDTLS_MPI_CHK(mbedtls_internal_ecp_init(grp));
    }

#endif /* MBEDTLS_ECP_INTERNAL_ALT */
    MBEDTLS_MPI_CHK(ecp_add_mixed(grp, R, &mP, R));
    MBEDTLS_MPI_CHK(ecp_normalize_jac(grp, R));

cleanup:

#if defined(MBEDTLS_ECP_INTERNAL_ALT)
    if (is_grp_capable) {
        mbedtls_internal_ecp_free(grp);
    }

#endif /* MBEDTLS_ECP_INTERNAL_ALT */
    mbedtls_ecp_point_free(&mP);

    return (ret);
}


#if defined(ECP_MONTGOMERY)
/*
 * Check validity of a public key for Montgomery curves with x-only schemes
 */
static int ecp_check_pubkey_mx(const mbedtls_ecp_group *grp, const mbedtls_ecp_point *pt) {
    /* [Curve25519 p. 5] Just check X is the correct number of bytes */
    /* Allow any public value, if it's too big then we'll just reduce it mod p
     * (RFC 7748 sec. 5 para. 3). */
    if (mbedtls_mpi_size(&pt->X) > (grp->nbits + 7) / 8)
        return (MBEDTLS_ERR_ECP_INVALID_KEY);

    return (0);
}
#endif /* ECP_MONTGOMERY */

/*
 * Check that a point is valid as a public key
 */
int mbedtls_ecp_check_pubkey(const mbedtls_ecp_group *grp, const mbedtls_ecp_point *pt) {
    /* Must use affine coordinates */
    if (mbedtls_mpi_cmp_int(&pt->Z, 1) != 0)
        return (MBEDTLS_ERR_ECP_INVALID_KEY);

#if defined(ECP_MONTGOMERY)
    if (ecp_get_type(grp) == ECP_TYPE_MONTGOMERY)
        return (ecp_check_pubkey_mx(grp, pt));
#endif
#if defined(ECP_SHORTWEIERSTRASS)
    if (ecp_get_type(grp) == ECP_TYPE_SHORT_WEIERSTRASS)
        return (ecp_check_pubkey_sw(grp, pt));
#endif
    return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);
}

/*
 * Check that an mbedtls_mpi is valid as a private key
 */
int mbedtls_ecp_check_privkey(const mbedtls_ecp_group *grp, const mbedtls_mpi *d) {
#if defined(ECP_MONTGOMERY)
    if (ecp_get_type(grp) == ECP_TYPE_MONTGOMERY) {
        /* see RFC 7748 sec. 5 para. 5 */
        if (mbedtls_mpi_get_bit(d, 0) != 0 ||
                mbedtls_mpi_get_bit(d, 1) != 0 ||
                mbedtls_mpi_bitlen(d) - 1 != grp->nbits)    /* mbedtls_mpi_bitlen is one-based! */
            return (MBEDTLS_ERR_ECP_INVALID_KEY);

        /* see [Curve25519] page 5 */
        if (grp->nbits == 254 && mbedtls_mpi_get_bit(d, 2) != 0)
            return (MBEDTLS_ERR_ECP_INVALID_KEY);

        return (0);
    }
#endif /* ECP_MONTGOMERY */
#if defined(ECP_SHORTWEIERSTRASS)
    if (ecp_get_type(grp) == ECP_TYPE_SHORT_WEIERSTRASS) {
        /* see SEC1 3.2 */
        if (mbedtls_mpi_cmp_int(d, 1) < 0 ||
                mbedtls_mpi_cmp_mpi(d, &grp->N) >= 0)
            return (MBEDTLS_ERR_ECP_INVALID_KEY);
        else
            return (0);
    }
#endif /* ECP_SHORTWEIERSTRASS */

    return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);
}

/*
 * Generate a keypair with configurable base point
 */
int mbedtls_ecp_gen_keypair_base(mbedtls_ecp_group *grp,
                                 const mbedtls_ecp_point *G,
                                 mbedtls_mpi *d, mbedtls_ecp_point *Q,
                                 int (*f_rng)(void *, unsigned char *, size_t),
                                 void *p_rng) {
    int ret;
    size_t n_size = (grp->nbits + 7) / 8;

#if defined(ECP_MONTGOMERY)
    if (ecp_get_type(grp) == ECP_TYPE_MONTGOMERY) {
        /* [M225] page 5 */
        size_t b;

        do {
            MBEDTLS_MPI_CHK(mbedtls_mpi_fill_random(d, n_size, f_rng, p_rng));
        } while (mbedtls_mpi_bitlen(d) == 0);

        /* Make sure the most significant bit is nbits */
        b = mbedtls_mpi_bitlen(d) - 1;   /* mbedtls_mpi_bitlen is one-based */
        if (b > grp->nbits)
            MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(d, b - grp->nbits));
        else
            MBEDTLS_MPI_CHK(mbedtls_mpi_set_bit(d, grp->nbits, 1));

        /* Make sure the last two bits are unset for Curve448, three bits for
           Curve25519 */
        MBEDTLS_MPI_CHK(mbedtls_mpi_set_bit(d, 0, 0));
        MBEDTLS_MPI_CHK(mbedtls_mpi_set_bit(d, 1, 0));
        if (grp->nbits == 254) {
            MBEDTLS_MPI_CHK(mbedtls_mpi_set_bit(d, 2, 0));
        }
    } else
#endif /* ECP_MONTGOMERY */
#if defined(ECP_SHORTWEIERSTRASS)
        if (ecp_get_type(grp) == ECP_TYPE_SHORT_WEIERSTRASS) {
            /* SEC1 3.2.1: Generate d such that 1 <= n < N */
            int count = 0;

            /*
             * Match the procedure given in RFC 6979 (deterministic ECDSA):
             * - use the same byte ordering;
             * - keep the leftmost nbits bits of the generated octet string;
             * - try until result is in the desired range.
             * This also avoids any biais, which is especially important for ECDSA.
             */
            do {
                MBEDTLS_MPI_CHK(mbedtls_mpi_fill_random(d, n_size, f_rng, p_rng));
                MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(d, 8 * n_size - grp->nbits));

                /*
                 * Each try has at worst a probability 1/2 of failing (the msb has
                 * a probability 1/2 of being 0, and then the result will be < N),
                 * so after 30 tries failure probability is a most 2**(-30).
                 *
                 * For most curves, 1 try is enough with overwhelming probability,
                 * since N starts with a lot of 1s in binary, but some curves
                 * such as secp224k1 are actually very close to the worst case.
                 */
                if (++count > 30)
                    return (MBEDTLS_ERR_ECP_RANDOM_FAILED);
            } while (mbedtls_mpi_cmp_int(d, 1) < 0 ||
                     mbedtls_mpi_cmp_mpi(d, &grp->N) >= 0);
        } else
#endif /* ECP_SHORTWEIERSTRASS */
            return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);

cleanup:
    if (ret != 0)
        return (ret);

    return (mbedtls_ecp_mul(grp, Q, d, G, f_rng, p_rng));
}

/*
 * Generate key pair, wrapper for conventional base point
 */
int mbedtls_ecp_gen_keypair(mbedtls_ecp_group *grp,
                            mbedtls_mpi *d, mbedtls_ecp_point *Q,
                            int (*f_rng)(void *, unsigned char *, size_t),
                            void *p_rng) {
    return (mbedtls_ecp_gen_keypair_base(grp, &grp->G, d, Q, f_rng, p_rng));
}

/*
 * Generate a keypair, prettier wrapper
 */
int mbedtls_ecp_gen_key(mbedtls_ecp_group_id grp_id, mbedtls_ecp_keypair *key,
                        int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) {
    int ret;

    if ((ret = mbedtls_ecp_group_load(&key->grp, grp_id)) != 0)
        return (ret);

    return (mbedtls_ecp_gen_keypair(&key->grp, &key->d, &key->Q, f_rng, p_rng));
}

/*
 * Check a public-private key pair
 */
int mbedtls_ecp_check_pub_priv(const mbedtls_ecp_keypair *pub, const mbedtls_ecp_keypair *prv) {
    int ret;
    mbedtls_ecp_point Q;
    mbedtls_ecp_group grp;

    if (pub->grp.id == MBEDTLS_ECP_DP_NONE ||
            pub->grp.id != prv->grp.id ||
            mbedtls_mpi_cmp_mpi(&pub->Q.X, &prv->Q.X) ||
            mbedtls_mpi_cmp_mpi(&pub->Q.Y, &prv->Q.Y) ||
            mbedtls_mpi_cmp_mpi(&pub->Q.Z, &prv->Q.Z)) {
        return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);
    }

    mbedtls_ecp_point_init(&Q);
    mbedtls_ecp_group_init(&grp);

    /* mbedtls_ecp_mul() needs a non-const group... */
    mbedtls_ecp_group_copy(&grp, &prv->grp);

    /* Also checks d is valid */
    MBEDTLS_MPI_CHK(mbedtls_ecp_mul(&grp, &Q, &prv->d, &prv->grp.G, NULL, NULL));

    if (mbedtls_mpi_cmp_mpi(&Q.X, &prv->Q.X) ||
            mbedtls_mpi_cmp_mpi(&Q.Y, &prv->Q.Y) ||
            mbedtls_mpi_cmp_mpi(&Q.Z, &prv->Q.Z)) {
        ret = MBEDTLS_ERR_ECP_BAD_INPUT_DATA;
        goto cleanup;
    }

cleanup:
    mbedtls_ecp_point_free(&Q);
    mbedtls_ecp_group_free(&grp);

    return (ret);
}

#if defined(MBEDTLS_SELF_TEST)

/*
 * Checkup routine
 */
int mbedtls_ecp_self_test(int verbose) {
    int ret;
    size_t i;
    mbedtls_ecp_group grp;
    mbedtls_ecp_point R, P;
    mbedtls_mpi m;
    unsigned long add_c_prev, dbl_c_prev, mul_c_prev;
    /* exponents especially adapted for secp192r1 */
    const char *exponents[] = {
        "000000000000000000000000000000000000000000000001", /* one */
        "FFFFFFFFFFFFFFFFFFFFFFFF99DEF836146BC9B1B4D22830", /* N - 1 */
        "5EA6F389A38B8BC81E767753B15AA5569E1782E30ABE7D25", /* random */
        "400000000000000000000000000000000000000000000000", /* one and zeros */
        "7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF", /* all ones */
        "555555555555555555555555555555555555555555555555", /* 101010... */
    };

    mbedtls_ecp_group_init(&grp);
    mbedtls_ecp_point_init(&R);
    mbedtls_ecp_point_init(&P);
    mbedtls_mpi_init(&m);

    /* Use secp192r1 if available, or any available curve */
#if defined(MBEDTLS_ECP_DP_SECP192R1_ENABLED)
    MBEDTLS_MPI_CHK(mbedtls_ecp_group_load(&grp, MBEDTLS_ECP_DP_SECP192R1));
#else
    MBEDTLS_MPI_CHK(mbedtls_ecp_group_load(&grp, mbedtls_ecp_curve_list()->grp_id));
#endif

    if (verbose != 0)
        mbedtls_printf("  ECP test #1 (constant op_count, base point G): ");

    /* Do a dummy multiplication first to trigger precomputation */
    MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&m, 2));
    MBEDTLS_MPI_CHK(mbedtls_ecp_mul(&grp, &P, &m, &grp.G, NULL, NULL));

    add_count = 0;
    dbl_count = 0;
    mul_count = 0;
    MBEDTLS_MPI_CHK(mbedtls_mpi_read_string(&m, 16, exponents[0]));
    MBEDTLS_MPI_CHK(mbedtls_ecp_mul(&grp, &R, &m, &grp.G, NULL, NULL));

    for (i = 1; i < sizeof(exponents) / sizeof(exponents[0]); i++) {
        add_c_prev = add_count;
        dbl_c_prev = dbl_count;
        mul_c_prev = mul_count;
        add_count = 0;
        dbl_count = 0;
        mul_count = 0;

        MBEDTLS_MPI_CHK(mbedtls_mpi_read_string(&m, 16, exponents[i]));
        MBEDTLS_MPI_CHK(mbedtls_ecp_mul(&grp, &R, &m, &grp.G, NULL, NULL));

        if (add_count != add_c_prev ||
                dbl_count != dbl_c_prev ||
                mul_count != mul_c_prev) {
            if (verbose != 0)
                mbedtls_printf("failed (%u)\n", (unsigned int) i);

            ret = 1;
            goto cleanup;
        }
    }

    if (verbose != 0)
        mbedtls_printf("passed\n");

    if (verbose != 0)
        mbedtls_printf("  ECP test #2 (constant op_count, other point): ");
    /* We computed P = 2G last time, use it */

    add_count = 0;
    dbl_count = 0;
    mul_count = 0;
    MBEDTLS_MPI_CHK(mbedtls_mpi_read_string(&m, 16, exponents[0]));
    MBEDTLS_MPI_CHK(mbedtls_ecp_mul(&grp, &R, &m, &P, NULL, NULL));

    for (i = 1; i < sizeof(exponents) / sizeof(exponents[0]); i++) {
        add_c_prev = add_count;
        dbl_c_prev = dbl_count;
        mul_c_prev = mul_count;
        add_count = 0;
        dbl_count = 0;
        mul_count = 0;

        MBEDTLS_MPI_CHK(mbedtls_mpi_read_string(&m, 16, exponents[i]));
        MBEDTLS_MPI_CHK(mbedtls_ecp_mul(&grp, &R, &m, &P, NULL, NULL));

        if (add_count != add_c_prev ||
                dbl_count != dbl_c_prev ||
                mul_count != mul_c_prev) {
            if (verbose != 0)
                mbedtls_printf("failed (%u)\n", (unsigned int) i);

            ret = 1;
            goto cleanup;
        }
    }

    if (verbose != 0)
        mbedtls_printf("passed\n");

cleanup:

    if (ret < 0 && verbose != 0)
        mbedtls_printf("Unexpected error, return code = %08X\n", ret);

    mbedtls_ecp_group_free(&grp);
    mbedtls_ecp_point_free(&R);
    mbedtls_ecp_point_free(&P);
    mbedtls_mpi_free(&m);

    if (verbose != 0)
        mbedtls_printf("\n");

    return (ret);
}

#endif /* MBEDTLS_SELF_TEST */

#endif /* !MBEDTLS_ECP_ALT */

#endif /* MBEDTLS_ECP_C */
